A Primer on Vorticity for

Application in Supercells and Tornadoes

Charles A. Doswell III

Cooperative Institute for Mesoscale Meteorological
Studies

Norman, Oklahoma

Last update: 27 November 2000... some minor cosmetic
revisions and clarifications, plus some substantial changes to
Mathematical Diversion - 4.

Disclaimer: This material is provided as a convenience to
World-Wide Web users interested in supercell thunderstorms and
tornadoes. It is not endorsed by anyone other than me; notably, it is
not any official document of MAT, MRAD, NSSL, ERL, OAR, NOAA, DoC, or
the President of the United States. This tutorial has benefitted from
comments by my colleague, Dr. Robert Davies-Jones, but that does not
imply that he endorses or underwrites these contents. That is, if
there are any errors or problems, they are mine alone. Some minor
changes/corrections have resulted from e-mails sent to me by Mike
Branick, Matt Bunkers, and Prof. Bob Terrell (at Cornell U.) ...
Thanks!

1. Introduction

Vorticity is a topic that can be plagued with confusion and
misunderstanding, although it has considerable power in helping to
understand the processes that go on in a vortex. Vorticity is a
property of the flow of air (or any other fluid, for that
matter). Air at rest has some temperature and dewpoint, but it has no
vorticity. Mere motion does not guarantee that the air has vorticity,
however. Moving fluids acquire properties that allow someone to
describe that flow in great detail. In order to develop a deep
understanding of these properties, you have to delve into the
discipline known as fluid dynamics . This subject can involve
quite sophisticated mathematics, but for the purposes of this primer,
I am going to try to keep the mathematics to a minimum. I am going to
include equations for the benefit of those who can use them, but I
hope to be able to achieve some success at explaining the basic
concepts even to those who know no mathematics at all. You can be the
judge of my success. Mathematical diversions can be skipped over by
the mathematically-challenged.

Vorticity is one of four quantities that are called the
kinematic properties of fluid flow. The wind flow is
described as a vector quantity; 2-dimensional vectors require
two numbers to describe them fully; 3-dimensional vectors need three
numbers, etc. Let us define the wind field as a function of position
in a coordinate system; for simplicity, let the coordinate system be
the standard 3-dimensional Cartesian system (x ,y
,z ), where (x ,y ) are the two horizontal
coordinates and z is the vertical coordinate. Thus, since the
wind vector V can vary according to its position in space, it
is also denoted as V(x ,y ,z ). Vectors
will be indicated in boldface characters. Let i, j, and
k denote unit vectors of position in the 3-dimensional
coordinate system (see Fig. 1),

Fig. 1. Illustration of the basic
definitions of the wind in a 3-dimensional Cartesian coordinate
system.

corresponding to the x , y , and z
directions, respectively. The wind vector V, therefore, has
three components: u , v , and w . We also can
describe V as Vh+wk, where
Vh is the horizontal wind vector and
wk is the vertical wind vector.

For the horizontal (2-dimensional) wind, then, it takes two
quantities to describe that vector. The mathematically correct way is
to use the components I have already described and illustrated.
However, horizontal wind also can be described in another way, that
might be more familiar: wind speed and wind direction . When
meteorologists describe the wind in this way, they refer to the
direction from which the wind is blowing. A wind from north
to south is called a northerly wind, for example. In
Fig. 2,

Fig. 2. An example showing two
ways to define the wind: by (u,v) components and by
speed and direction.

the components are u = 3, v = 4. This means that the
speed of the wind is V = 5. The direction is shown as a ~ 37 deg ... but this is not the way a
meteorologist would refer to it. Instead, the meteorological
wind direction is described as b = 180 deg
+ a = 217 deg ... the wind is from the
south-southwest. If 180 deg + a is greater
than 360 deg (say a = 280 deg) then the
meteorological wind direction b = a - 180 deg = 100 deg. As shown in Fig. 2, the
wind speed and direction are related directly to the u - and
v
-components.[1]

3. Understanding the Vector Character of Vorticity

Vorticity is a 3-dimensional vector, just like the wind
field. It is associated with the spatial variation of the wind field;
the wind changes from point to point in space, but not all such
changes produce vorticity. Since the 3-d vorticity is a vector, it
has components, just like the wind: If w is the 3-d vorticity vector then it has
three Cartesian coordinate components (x,h,z). Each of the three components
represents the vorticity associated with flow in a 2-d plane normal
to the component. The x-component of the vorticity (x) is perpendicular to the y-z plane, the
y-component of the vorticity (h) is
perpendicular to the x-z plane, and the z-component of the vorticity
(z) is perpendicular to the x-y
plane. Positive values of the vorticity components follow the
right-hand rule (see below). Details of this are found in
Mathematical Diversion -1. Meteorologists,
in general, are most accustomed to talking and thinking about the
vertical component of vorticity (z).

Vorticity in its 2-dimensional forms results from two different
types of wind variation: curvature of the flow, and differences in
wind speed in a direction perpendicular to the flow. When looking at
a 2-d vortex, it is easy imagine how the curvature of the flow
produces vorticity, as in Fig.
3.

Fig. 3. An illustration of
circular flow, showing the vorticity associated with this flow. The
thin lines show the flow direction and the bold line indicates the
sense of rotation associated with the vorticity. It is assumed that
there is no shear vorticity in this flow, except at the exact center
of the circular flow, where there is infinite shear (the
flow must go to zero at r=0, implying an
infinite vorticity).

Fig. 4. An illustration of
straight flow (no curvature), but with speed shear, showing the
vorticity associated with this flow. As in Fig. 3, the thin lines
show the flow direction and the bold line indicates the sense of
rotation associated with the vorticity.

variations in wind speed can also produce vorticity. The figure
indicates the sort of rotation that is implied by this form of wind
shear, but the fluid need not actually be rotating this way. Fluids
do not behave as solids do ... imagine rolling a pencil in between
the palms of your hand. Since a pencil is a solid object, it can only
rotate between your hands. However, fluids have another
"option" ... to see this, imagine pushing on a book that has been
glued to the top of a table. What happens is shown in
Fig. 5.

Fig. 5. Schematic showing the
shearing action associated with a book, glued to a table top,
subjected to a shearing force.

The pages of the book can slide past each other and the book is
"sheared" without rotation. In a very similar way, the layers of a
fluid can slide past each other when subjected to shearing forces.
This produces vorticity in the flow, but without closed
rotation!

Now, picture a 2-d Cartesian coordinate system that is set up such
that one axis is aligned with the flow at some point in the fluid,
and the other axis is perpendicular to the first. In meteorology,
this is referred to as "natural" coordinates. The two different
contributions to vorticity are easily seen in natural coordinates.
The shearing contribution is shown in
Fig. 6a, whereby vorticity arises
from changes in wind speed V in the direction perpendicular to
the flow (at the central point). The curvature contribution shown in
Fig. 6b is derived from changes in
wind direction a along the flow (at the central point).

Fig. 6. Schematic in natural
coordinates, defined at the point indicated by the small circle at
the base of the dark vector; the s-direction is always pointed in the direction of the
flow and then-direction is pointed
to the right and perpendicular to the s-direction. In (a),
the flow is straight, but speed increases to the right; in (b) the
flow is curved but the speed is not changing in the direction of the
flow.

It is most common for the 2-d flow to have both shear and
curvature at the same time, so it can happen that the contribution to
2-d vorticity from shear is of the opposite sign as the
contribution from curvature, just as it is possible that they have
the same sign. In fact, as a special case, the flow can form a
closed circle even though the total vorticity is zero! This happens
when the shear term is of the opposite sign to, and is exactly the
same magnitude as the curvature term. Even stranger, if the shear
magnitude is large enough, the total vorticity could have the
opposite sign to the curvature vorticity! Imagine a Rankine Combined
Vortex (see Fig. 8, below). It is
discussed in Mathematical Diversion - 2
that outside of the so-called Radius of Maximum Wind
(ro) in the Rankine Combined Vortex, the tangential
velocity decreases in proportion to 1/r, where r=the
radius. The total vorticity outside of ro is zero
because when tangential velocity decreases at that rate, the
shear vorticity is of opposite sign to the curvature
vorticity, and it exactly cancels the curvature vorticity. If the
tangential velocity decreases with radius even faster than
1/r, then the vorticity outside of ro
becomes negative! Thus, it's possible for a vortex to be rotating
cyclonically throughout, with a cyclonic vorticity core and a ring of
anticyclonic vorticity surrounding it. In fact, this is
actually observed in the atmosphere in association with tornadoes and
in the outflow region (aloft) of hurricanes.

For 2-d flow (say, at the surface), the vorticity vector points
either up or down, perpendicular to the surface. The voriticity sign
convention follows what is called the "right-hand rule": hold the
right hand on the paper with the thumb in the "hitchhike" position;
if the fingers on the right hand curve in the direction associated
with the vorticity (as in Figs. 3 and 4), then when the thumb is
pointing up, the vorticity is positive. If the fingers curving in the
sense of rotation force the thumb to point downward, the vorticity is
negative.

It is relatively easy to extend this concept to three dimensions
... the 3-dimensional vorticity vector includes contributions from
three mutually perpendicular surfaces: a horizontal surface (x , y
), and two vertical surfaces (x , z ) and (y , z ).
The total 3-dimensional vorticity vector is the vector sum of these
three. It also follows the same right-hand rule: if the right-hand's
thumb points in the direction of the vorticity vector, the fingers
curl about that thumb in a way that shows the sense of rotation
produced by that 3-dimensional vorticity.

Consider the following example of the vertical component of
vorticity (z), shown in
Fig. 7.

Fig. 7. A horizontal flow field,
with the winds indicated by meteorological "wind barbs" instead of
vectors. A positive vertical component of vorticity
(z) is indicated by
the stippling, getting larger as the stippling gets denser; the lines
with arrows are subjectively-drawn streamlines (see
discussion below).

Wind barbs are shown instead of wind vectors ... the notation is
conventional. For this system, it can be seen that the peak vorticity
is to the southwest of the center of circulation, where the curvature
is high and the shear contribution to the vorticity is of the same
sign as the curvature. Flow streamlines have been sketched in by hand
... streamlines are defined to be lines that are everywhere
parallel to the wind(strictly speaking, they are
everywhere tangential to the wind flow). Where the streamlines
have high curvature, the vorticity is high owing to the curvature
contribution. For instance, note the zone wrapped around the vortex
where the streamlines make an abrupt turn; this coincides with a long
zone of high vorticity (along a trailing cold front-like structure).
On the other hand, to the northeast of the center of circulation
(near the edge of the figure), there is a region of high vorticity
due to the shear contribution. For these particular data, the shear
term by itself does not create very much high vorticity, unless it
coincides with contributions from the curvature term, whereas there
are several vorticity maxima associated with strong flow curvature.
This is not always the case, but depends on the flow situation under
consideration.

5. Curvature and Relative Flow

A common problem in understanding the role of vorticity in flow
curvature is that the vorticity field is what is called a Galilean
invariant property of the flow. Simply put, Galilean invariance
means that it you can change to any coordinate system moving with a
constant velocity and it does not change the vorticity. However, the
partitioning of the contributions to vorticity from shear and
curvature is not a Galilean invariant property (see Viudez and
Haney 1996). The amount of curvature we see in the flow can depend on
the motion of the coordinate system.

To see this, consider the following simple example. Suppose we
have a Rankine Combined Vortex, as shown in
Fig. 8.

Fig. 8. Schematic showing the
effect of adding increasingly strong constant background flows to a
Rankine vortex. The vortex core is shown by the stippling, the blue
arrows indicate the vortex flow, the red arrows denote the background
flow, the green arrows show the sum of the vortex and the background
winds, and the streamlines show the combined flow.

In this vortex, the flow is perfectly symmetric and circular, with
contributions from both shear and curvature. However, in Fig. 8b, a
background flow (that is constant everywhere) is added to the vortex,
with that background flow being pure westerly with a speed equal to
the speed at the radius of maximum winds in the pure vortex flow.
Streamlines have been added to indicate the flow of the sum of these
two. In Fig. 8c, the background westerly flow speed is twice that of
that of the maximum vortex flow, whereas in Fig. 8d, the background
westerly flow is three times that of the maximum vortex flow. If
considered in a coordinate system traveling with the
background flow in Figs. 8b-d, all the flows would be identical to
that of Fig. 8a. However, it can be seen that by adding this
background flow, the curvature of the resultant flow in a fixed
coordinate system (i.e., not moving with the background flow)
decreases as this background flow increases. In all cases, the
vorticity remains the same, because the background flow has
no vorticity!

In situations with mesocyclones as seen on a Doppler
radar,[2] it is
common for the flow in a ground-relative sense to be very asymmetric
(perhaps with stronger inbound flow than outbond flow or vice-versa),
because the vortex is moving. For a moving vortex, the flow on the
side of the vortex that is going in the direction of storm motion,
its ground-relative flow is the sum of the vortex flow and its
motion. On the opposite side of the vortex, the flow is the
difference between the vortex flow and the motion, so it is
relatively slower. In a coordinate system moving with the
vortex, the vortex flow typically is much more symmetric.

6. Streamwise and Crosswise Vorticity

I already have defined streamlines to be lines everywhere tangent
to (i.e., parallel to) the wind flow. It is important to remember
that most meteorologists are accustomed to dealing with is the
vertical component of the vector vorticity. However, the
vertical component of the vector vorticity typically is the
smallest of the three
components.[3]
That is, the variation of wind with height usually is much larger
than the variation of wind along a more or less level surface (like
on a surface of constant pressure). The horizontal part of the
vorticity has two components, just like the horizontal wind has two
components. Like the wind vector, the 3-dimensional vorticity vector
is dominated by its horizontal part whenever the flow field can be
said to be in near-hydrostatic balance (as discussed in the
Mathematical Diversion - 1). Therefore, it
should be relatively easy to imagine a field of vorticity vectors
that might look very similar to a field of wind vectors, but which in
fact are horizontal vorticity vectors wh (which have both x- and
y-components) derived from the wind. As shown in
Mathematical Diversion - 3, the horizontal
vorticity vectors are dominated by the vertical wind shear of the
horizontal wind. In fact, under the hydrostatic assumption, the
horizontal vorticity vector is very nearly perpendicular to the
vertical wind shear vector. However, there is no guarantee what the
relationship is between the direction of the horizontal wind
vector on some surface and that of the associated horizontal
vorticity vector, which depends on the vertical shear
vector of the horizontal wind.

When the horizontal vorticity vector and the horizontal wind
vector are parallel, the horizontal vorticity is said to be
streamwise , and when they are perpendicular, the horizontal
vorticity is said to be crosswise . When the horizontal wind
and vorticity vectors are parallel but point in opposite directions
(antiparallel), the horizontal vorticity is said to be
antistreamwise .

8. Vortex Lines

As noted in
Doswell
(1991), the notions of helicity can be developed from
this point rather easily. That is not the point of these
notes. Rather, I now want to move on to the idea of vortex
lines. If streamlines are everywhere tangent to the velocity
vectors, it seems straightforward to draw upon this analogy and
picture lines that are everywhere tangent to the vorticity
vectors. From a strictly two-dimensional viewpoint, this is
moderately interesting, but the real value comes from the realization
that there also is a vertical component to the vorticity vector.

In the same way that streamlines point in the direction of the
wind flow, the vortex lines point in the direction of the vorticity
vectors. The same old "right-hand rule" applies: if the right-hand
thumb points in the direction of the vortex line, the sense of
rotation is given by the fingers of the right hand.

When it comes to tornadoes and mesocyclones in storms, we are
interested in rotation about a vertical axis. That is, the
vertical component of the vorticity is what's important even though,
on large scales associated with a thunderstorm's environment,
the horizontal component is larger. Mesocyclones and tornadoes are
not synoptic-scale! Recall that I have suggested that the vertical
component is often much smaller than the horizontal component in
situations where hydrostatic balance
applies.[4] If
we make the approximation that the storm inflow and updraft simply
acts on the environmental horizontal vorticity, an
interesting picture begins to emerge.

If the originally horizontal vortex lines can be tilted into the
vertical by some mechanism, then the vortex lines develop a vertical
component, and so the vertical component of vorticity can be changed
simply by tilting of the existing purely horizontal vorticity. It is
widely accepted that this tilting of horizonal vorticity is the
primary mechanism for creating mid-level mesocyclones in supercells
(see Brooks et al. 1994)

Suppose the flow is such that the low-level wind shear is
producing only crosswise vorticity. For example, a pure change of
wind speed with height (wind direction remains constant) produces
only crosswise vorticity. This is analogous to the case shown in Fig.
5 with the sheared book. Vortex lines are all horizontal in the
environment.[5]
Such a case is depicted in Fig. 9.

Fig. 9. Schematic showing the
development of counter-rotating vortices on either side of an updraft
from tilting the vortex lines associated with pure
crosswise
vorticity. Notice the sense of
rotation indicated by the circles around the vortex lines, following
the right-hand rule. This schematic is in the Northern hemisphere,
where positive vertical vorticity is cyclonic , indicated by
the "C" and negative vertical vorticity is anticyclonic ,
indicated by the "A".

On either side of the updraft, the vortex lines are tilted upward
by the updraft, such that on one side of the updraft, a cyclonically
rotating vertical component of vorticity is created. On the opposite
side of the updraft, of course, an anticyclonically rotating vertical
component of vorticity is found.

What happens when the vorticity is purely streamwise? This is
illustrated in Fig. 10;

Fig. 10. As in Fig. 9, except that
it is a schematic showing the development of a cyclonically rotating
updraft from tilting the vortex lines associated with pure
streamwise
vorticity.

rather than producing a cyclonic-anticyclonic couplet of vortices
flanking the updraft, what one finds is a helically rotating updraft.
Note the difference between Fig. 10 and Fig. 9: the vortex line is
shown going up in Fig. 10 but not coming down ... this is an
important difference between streamwise and crosswise tilting
situations. The vortex line brought up in the updraft in the
streamwise case eventually turns downward somewhere
downstream, but this is not shown in Fig. 10. It turns out that if we
define some area on a horizontal plane, the vorticity about the
vertical in that plane is related to the number of vortex lines
inside that area. Thus, the more densely packed the vortex lines
passing through some plane, the greater the vorticity. This is
illustrated in Fig. 11, where the
vortex lines are rather densely packed in one region as they pass
through the plane.

This can have some interesting consequences in trying to picture
the vortex line structures ... note also that in viscous fluids
(where the velocity goes to zero at any solid surface), the vorticity
component normal to the surface must vanish, but the flow can still
have horizontal vorticity at the surface (see Warsi 1993; p. 22).
Thus, a vortex line approaching the surface in a viscous fluid must
become increasingly horizontal. There are no other options. This law
has some pretty obvious implications for atmospheric vortices, such
as mesocyclones and tornadoes. What we see as a funnel aloft (or a
Tornado Vortex Signature [TVS] on a Doppler radar) does not have
vortex lines that begin and end somewhere in the air ... since that
is not allowed, what must be happening is shown in
Fig. 13.

The vortex lines that cluster somewhere aloft must diverge above
and below the intense vortex, either ending at the surface or forming
closed loops. The figure shows the vortex lines ending at the
tropopause, but they really extend beyond it ... perhaps in complex
ways.

What we call "touchdown" of a tornado is actually an
intensification of the vortex at the ground, which can be viewed
as a clustering of the vortex lines. Nothing material is actually
coming down as the tornado "touches down" ... this process might more
properly called a "spin-up" or "surface intensification." The intense
part of a vortex aloft can build both upward and downward, as shown
in Fig. 14.

Fig. 14. As in Fig. 13, except
that the vortex has intensified both upward and downward and now is a
tornado.

Apparently, tornadic vortices can intensify first at the surface
(Fig. 15) and then build upward.

Fig. 15. Schematic showing vortex
lines in a situation where the vortex intensifies first at low
levels.

After some time period, Fig. 15 could evolve into something
resembling Fig. 14; that is, it also can evolve into a tornado
through a deep layer. Thus, the intensified vortex that we call a
tornado might be seen to develop upward from the surface, downward
from aloft, or both upward and downward.